Greeble Cube Microdisplacement - how to prevent edges from opening up and becoming black
I can't replicate your problem with Blender 2.79.6 as I don't have access to 2.79.6 and I can't load your file into 2.79.5 since I don't have a 'safe' environment that I can risk loading "foreign" blend files). I did try loading into 2.79b but without success (from your file I couldn’t get any displacement working - even after tweaking the material to remove the unsupported nodes).

I tried replicating your situation afresh in 2.79.5 using the default cube, adding Subsurface modifiers (one set to '6' and the other set to '3' - so 9 subdivisions) and with the following material :



This produced the following result :



I don't see any dark gaps and the edges all seem to displace correctly without splitting.

Can you try following these steps in a fresh file to see if you get the same results? - or identify anything that you have done different that may have resulted in the issue.